12 men can do a piece of work in 24 days. How many days are needed to complete the work, if 8 men do this work?

### Concept & Formula This is a direct application of the fundamental inverse proportion rule (Chain Rule) between the number of workers and the time taken to complete a constant amount of work: $$ M_1 \times D_1 = M_2 \times D_2 $$ ### Step-by-Step Solution 1. **Identify the variables:** $M_1 = 12$ men $D_1 = 24$ days $M_2 = 8$ men $D_2 = ?$ 2. **Apply the formula:** $$ 12 \times 24 = 8 \times D_2 $$ $$ D_2 = \frac{12 \times 24}{8} $$ $$ D_2 = 12 \times 3 = 36 $$ ### Exam Strategy & Shortcut Recognize the ratio of workers. The workforce is reduced from 12 to 8, which is a ratio of $2/3$. Time taken is inversely proportional, so multiply the original time by $3/2$. $24 \times (3/2) = 36$. ### Common Pitfall Accidentally setting up a direct proportion (cross-multiplying) instead of an inverse proportion, which would yield an illogical answer smaller than 24. ### Final Answer Therefore, the correct answer is **36**.